Lifting of nearly extraordinary Galois representations

Let be a continuous, 2-dimensional and absolutely irreducible mod p representation of the absolute Galois group of a number field K/Q. In this work, we study the existence of lifts of to GL2(Zp), for which the restrictions to the decomposition groups above p are abelian. The tools and philosophy come from the Taylor–Ramakrishna method. As an application we produce finitely ramified extensions over Q(μp∞) with Galois group SL2(Zp), for some p. These extensions are unramified at p.